IPACS Electronic library

Viscous Flows in a Half Space caused by
Tangential Vibrations on its Boundary.

Vladimir A. Vladimirov

The paper is devoted to the studies of viscous flows caused by a
vibrating boundary. The fluid domain is a half space, its boundary
is a non-deformable plane that exhibits purely tangential
vibrations. Such a simple geometrical setting allows us to study
general boundary velocity fields and to obtain general results. From
a practical viewpoint such boundary conditions may be seen as the
tangential vibrations of the material points of a plane stretchable
membrane. In contrast to the classical boundary layer theory we aim to build a global solution. In order to achieve this goal we employ the Vishik-Lyusternik approach combined with two-timing and averaging methods. Our main result is: we obtain a uniformly valid in the whole fluid domain approximation to the global solutions.
This solution corresponds to general boundary conditions and to
three different settings of the main small parameter. Our solution
always include the `inner' part and `outer' part that contain both
oscillating and non-oscillating components. It is shown that the
non-oscillating `outer' part of the solution is governed by the
either full Navier-Stokes equations or the Stokes equations (both
with the unit viscosity) and can be interpreted as a steady or
unsteady streaming. In contrast to the existing theories of a steady
streaming our solutions do not contain any secular (infinitely
growing with the inner normal coordinate) terms. The examples of the spatially periodic vibrations of the boundary and the angular
torsional vibrations of an infinite rigid disc are considered. These
examples are still brief and illustrative, while the core of the
paper is devoted to the adapting of the Vishik-Lyusternik method to
the development of the general theory of vibrational boundary
layers.